Quantized Repetitions of the Cuprate Pseudogap Line

The cuprate superconductors display several characteristic temperatures which decrease as the material composition is doped, tracing lines across the temperature-doping phase diagram. Foremost among these is the pseudogap transition. At a higher temperature a peak is seen in the magnetic susceptibility, and changes in symmetry and in transport are seen at other characteristic temperatures. We report a meta-analysis of all measurements of characteristic temperatures well above $T_c$ in strontium doped lanthanum cuprate (LSCO) and oxygen doped YBCO. The experimental corpus shows that the pseudogap line is one of a family of four straight lines which stretches across the phase diagram from low to high doping, and from $T_c$ up to $700$ K. These lines all originate from a single point near the overdoped limit of the superconducting phase and increase as doping is reduced. The slope of the pseudogap lines is quantized, with the second, third, and fourth lines having slopes that are respectively $1/2,\;1/3,$ and $1/4$ of the slope of the highest line. This pattern suggests that the cuprates host a single mother phase controlled by a 2-D sheet density which is largest at zero doping and which decreases linearly with hole density, and that the pseudogap lines, charge density wave order, and superconductivity are all subsidiary effects supported by the mother phase.